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43
GossipBased Computation of Aggregate Information
, 2003
"... between computers, and a resulting paradigm shift from centralized to highly distributed systems. With massive scale also comes massive instability, as node and link failures become the norm rather than the exception. For such highly volatile systems, decentralized gossipbased protocols are emergin ..."
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Cited by 297 (1 self)
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between computers, and a resulting paradigm shift from centralized to highly distributed systems. With massive scale also comes massive instability, as node and link failures become the norm rather than the exception. For such highly volatile systems, decentralized gossipbased protocols are emerging as an approach to maintaining simplicity and scalability while achieving faulttolerant information dissemination.
A Decentralized Algorithm for Spectral Analysis
, 2004
"... In many large network settings, such as computer networks, social networks, or hyperlinked text documents, much information can be obtained from the network’s spectral properties. However, traditional centralized approaches for computing eigenvectors struggle with at least two obstacles: the data ma ..."
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Cited by 53 (1 self)
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In many large network settings, such as computer networks, social networks, or hyperlinked text documents, much information can be obtained from the network’s spectral properties. However, traditional centralized approaches for computing eigenvectors struggle with at least two obstacles: the data may be difficult to obtain (both due to technical reasons and because of privacy concerns), and the sheer size of the networks makes the computation expensive. A decentralized, distributed algorithm addresses both of these obstacles: it utilizes the computational power of all nodes in the network and their ability to communicate, thus speeding up the computation with the network size. And as each node knows its incident edges, the data collection problem is avoided as well. Our main result is a simple decentralized algorithm for computing the top k eigenvectors of a symmetric weighted adjacency matrix, and a proof that it converges essentially in O(τmix log 2 n) rounds of communication and computation, where τmix is the mixing time of a random walk on the network. An additional contribution of our work is a decentralized way of actually detecting convergence, and diagnosing the current error. Our protocol scales well, in that the amount of computation performed at any node in any one round, and the sizes of messages sent, depend polynomially on k, but not at all on the (typically much larger) number n of nodes.
Symmetry analysis of reversible markov chains
 Internet Mathematics
, 2005
"... We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a selfadjoint operator with criteria for an eigenvector to descend to ..."
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Cited by 33 (11 self)
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We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a selfadjoint operator with criteria for an eigenvector to descend to an orbit graph. As examples, we show that the Metropolis construction can dominate a maxdegree construction by an arbitrary amount and that, in turn, the fastest mixing Markov chain can dominate the Metropolis construction by an arbitrary amount. 1
Asymptotic enumeration of spanning trees
 Combin. Probab. Comput
, 2005
"... Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that ..."
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Cited by 29 (6 self)
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Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call “tree entropy”, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasitransitive amenable graphs, extending a result of Burton and Pemantle (1993). §1. Introduction. Methods of enumeration of spanning trees in a finite graph G and relations to various areas of mathematics and physics have been investigated for more than 150 years. The number of spanning trees is often called the complexity of the graph, denoted here by τ(G). The best known formula for the complexity, proved in every basic text on graph
Entropy and the law of small numbers
 IEEE Trans. Inform. Theory
, 2005
"... Two new informationtheoretic methods are introduced for establishing Poisson approximation inequalities. First, using only elementary informationtheoretic techniques it is shown that, when Sn = �n i=1 Xi is the sum of the (possibly dependent) binary random variables X1, X2,..., Xn, with E(Xi) = p ..."
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Cited by 29 (11 self)
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Two new informationtheoretic methods are introduced for establishing Poisson approximation inequalities. First, using only elementary informationtheoretic techniques it is shown that, when Sn = �n i=1 Xi is the sum of the (possibly dependent) binary random variables X1, X2,..., Xn, with E(Xi) = pi and E(Sn) = λ, then D(PSn�Po(λ)) ≤ n� i=1 p 2 i + � n � i=1 H(Xi) − H(X1, X2,..., Xn), where D(PSn�Po(λ)) is the relative entropy between the distribution of Sn and the Poisson(λ) distribution. The first term in this bound measures the individual smallness of the Xi and the second term measures their dependence. A general method is outlined for obtaining corresponding bounds when approximating the distribution of a sum of general discrete random variables by an infinitely divisible distribution. Second, in the particular case when the Xi are independent, the following sharper bound is established,
Random walks on finite groups
 Encyclopaedia of Mathematical Sciences
, 2004
"... Summary. Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time ” of the chain, that is, the time ..."
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Cited by 20 (2 self)
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Summary. Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time ” of the chain, that is, the time at which the chain gives a good approximation of the limit distribution. A remarkable phenomenon known as the cutoff phenomenon asserts that this often happens abruptly so that it really makes sense to talk about “the mixing time”. Random walks on finite groups generalize card shuffling models by replacing the symmetric group by other finite groups. One then would like to understand how the structure of a particular class of groups relates to the mixing time of natural random walks on those groups. It turns out that this is an extremely rich problem which is very far to be understood. Techniques from a great
Mixing times of the biased card shuffling and the asymmetric exclusion process
 Trans. Amer. Math. Soc
, 2005
"... Abstract. Consider the following method of card shuffling. Start with a deck of N cards numbered 1 through N. Fix a parameter p between 0 and 1. In this model a “shuffle ” consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability p. If the coin ..."
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Cited by 18 (2 self)
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Abstract. Consider the following method of card shuffling. Start with a deck of N cards numbered 1 through N. Fix a parameter p between 0 and 1. In this model a “shuffle ” consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability p. If the coin comes up heads, then we arrange the two cards so that the lowernumbered card comes before the highernumbered card. If the coin comes up tails, then we arrange the cards with the highernumbered card first. In this paper we prove that for all p � = 1/2, the mixing time of this card shuffling is O(N 2), as conjectured by Diaconis and Ram (2000). Our result is a rare case of an exact estimate for the convergence rate of the Metropolis algorithm. A novel feature of our proof is that the analysis of an infinite (asymmetric exclusion) process plays an essential role in bounding the mixing time of a finite process. 1.
THE MARKOV CHAIN MONTE CARLO REVOLUTION
"... Abstract. The use of simulation for highdimensional intractable computations has revolutionized applied mathematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through microlocal analysis. 1. ..."
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Cited by 18 (1 self)
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Abstract. The use of simulation for highdimensional intractable computations has revolutionized applied mathematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through microlocal analysis. 1.
Separation cutoffs for birth death chains
, 2006
"... This paper gives a necessary and sufficient condition for a sequence of birth and death chains to converge abruptly to stationarity, that is, to present a cutoff. The condition involves the notions of spectral gap and mixing time. Y. Peres has observed that for many families of Markov chains, there ..."
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Cited by 12 (3 self)
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This paper gives a necessary and sufficient condition for a sequence of birth and death chains to converge abruptly to stationarity, that is, to present a cutoff. The condition involves the notions of spectral gap and mixing time. Y. Peres has observed that for many families of Markov chains, there is a cutoff if and only if the product of spectral gap and mixing time tends to infinity. We establish this for arbitrary birth and death chains in continuous time when the convergence is measured in separation and the chains all start at 0.